Table Of ContentSpringer Series in Computational Mathematics 59
Lehel Banjai
Francisco-Javier Sayas
Integral Equation
Methods
for Evolutionary
PDE
A Convolution Quadrature Approach
Springer Series in Computational Mathematics
Volume 59
SeriesEditors
RandolphE.Bank,DepartmentofMathematics,UniversityofCalifornia,
SanDiego,LaJolla,CA,USA
WolfgangHackbusch,Max-Planck-InstitutfürMathematikinden
Naturwissenschaften,Leipzig,Germany
JosefStoer,InstitutfürMathematik,UniversityofWürzburg,Würzburg,Germany
HarryYserentant,InstitutfürMathematik,TechnischeUniversitätBerlin,Berlin,
Germany
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Lehel Banjai (cid:129) Francisco-Javier Sayas
Integral Equation Methods
for Evolutionary PDE
A Convolution Quadrature Approach
LehelBanjai Francisco-JavierSayas
MaxwellInstituteforMathematical DepartmentofMathematicalSciences
Sciences UniversityofDelaware
Heriot-WattUniversity Newark,DE,USA
Edinburgh,UK
ThisworkwassupportedbyLondonMathematicalSociety,ResearchinPairs-Scheme4
ISSN0179-3632 ISSN2198-3712 (electronic)
SpringerSeriesinComputationalMathematics
ISBN978-3-031-13219-3 ISBN978-3-031-13220-9 (eBook)
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Dedicatedtoour families,closeand
extended.
Preface
The aim of this book is to serve as an introduction to both the theory of time-
domain boundary integral equations (TDBIE) and the analysis of their numerical
discretisationbyconvolutionquadratureandtheGalerkinboundaryelementmethod
(BEM). This includes describing the main algorithms, giving an indication of the
accelerationpossiblebytheuseofdatasparsemethods,andthroughoutdescribing
someexperiencesoftheauthorsfromyearsofworkinginthisareaandgivingsome
practicaladvice.Theemphasisofthebookisonhyperbolicproblems,specifically
acousticscatteringandtoamuchlesserextentscatteringofelectromagneticwaves.
Dueto somerecentinterestofthe firstauthor,we havealso includeda chapteron
fast methods for parabolic problems. One of the main motivations behind writing
thisbookhasbeentogobeyondthestandardsound–softandsound–hardexamples
ofscatteringbyobstacles.Therefore,wealsodescribeapplicationsinvolvingnon-
linearabsorbingboundaryconditionsandFEM/BEMcoupling.Anotherimportant
motivationwastoprovideshortMatlabcodesthatimplementthemainalgorithms
and illustrate the numerical performance of the schemes. The codes can be
downloadedfromthededicatedGitHubpage
https://github.com/lehelb/TDBIE-CQ-book
wheretranslationsintoPythonandfurthercodesandexamplescanbefound.Errata,
whentheycomeastheysurelywill,willalsobepostedthere.
While this is envisioned as a graduate level text, the prerequisites for fully
understandingthe textare notfew.Nevertheless,mostofthe backgroundrequired
shouldbe partof a solid undergraduate+masterlevelmathematicaleducation,and
the appendicesshould serve to refresh this knowledge.One clear exceptionis the
theoryofsteady-stateboundaryintegralequationsandtheirnumericaldiscretisation
by the boundary element method. Giving an introduction to this complex theory
and practical implementation would have changed the nature of this book (and
its length!).Fortunately,a numberof excellenttextbooksare now available in the
literaturethatwerefertothroughoutourtext.
As for other books on numerical methods for TDBIE, let us first mention
the earlier book by the second author [104]. While there are naturally many
vii
viii Preface
similaritiestothistext,wesignificantlyextendtheapplicationsweaddressandmore
stronglyemphasisepracticalaspectssuchasalgorithmsandqualitativebehaviourof
numericalsolutions,andincludeachapteronparabolicproblemsanddescriptionof
data-sparsetechniquesandmodifiedschemes.Themaindifferenceto[104]isthat
theanalysisinthecurrentbookisdonethroughoutviatheLaplacedomainwhereas
oneofthemainobjectivesoftheearlierbookwasthedevelopmentofatime-domain
programme.Whilefinishingthisbook,PaulMartin’sbook[88]on“Time-Domain
Scattering”appeared.Itcontainsanintroductiontotime-domainintegralequations
and also includes a section on convolution quadrature. The emphasis is however
quitedifferent,andthebooksnicelycomplementeachother.
Next, we give a brief description of the content of the book by describing the
topicscoveredbytheninechaptersandtheoneappendix.
1. Chapter1givesaninformalintroductiontocausalconvolutionswithaplethora
of examples. These range from fractional integrals/derivatives to a 3D scat-
tering problem solved by boundary integral equations. While descriptive and
introductory,thischapterisessentialfortheunderstandingoftheremainderof
thebook.
2. Chapter 2 introduces convolution quadrature (CQ) for hyperbolic symbols
based on linear multistep methods and gives the full convergenceanalysis. It
also describes a number of properties of CQ that will be important for the
applications we have in mind, e.g., positivity conservation. Generalised CQ
allowing for non-uniform time-steps and its analysis are also described. The
chapterendswithtwosectionsthatgivetheresultsneededtoanalyseGalerkin
discretisationsinthecasethatthehyperbolicsymbolK isanintegraloperator
as well as the analysis of the perturbations arising from the unavoidable
numericalquadrature.
3. Chapter 3 contains the main algorithms required for the implementation of
CQ, again with a view to applying these algorithms to integral operators
that are expensive to evaluate. Matlab implementation and simple numerical
experimentsarealsotobefoundinthechapter.
4. Chapter 4 does many things. It describes time-domain boundary integral
operatorsforthescalar acousticscatteringapplications.Theiranalysisviathe
Laplace domain based on the theory of Bamberger and Ha-Duong is given
in fullincludingcontinuityand coercivityestimatesfor the main operatorsas
wellasfortheCalderónoperator.Finally,thistheoryiscombinedwiththeCQ
theoryto describe the analysis of fully discrete schemes. Scattering problems
investigated include sound–hard and sound–soft scattering, and scattering by
obstaclessatisfyinglinearandnon-linearimpedanceboundaryconditions.
5. Relativelyshort,Chap.5introducesmultistageCQ basedonA-stableRunge-
Kuttamethods.Theconvergenceanalysisisgivenandorderreductionempha-
sised. Sound–softscattering is revisited as an example of the use of RK-CQ.
ShortMatlabcodesareagainprovided.
6. Chapter 6 describes the extension of the applicability of TDBIE+CQ to
Maxwellequations.Oncethe toolsandnotationisset, theanalysisis surpris-
Preface ix
inglysimilartotheacousticcase.Thechapter,however,endswithsomenotes
onthedifferencesbetweenthebehavioursofthenumericalschemesinthetwo
applicationareas.
7. To extend the applicability of TDBIE, it is important to be able to couple
withthemoreflexiblefiniteelementmethod(FEM)onboundeddomains.This
leadsto FEM-BEM couplingin the time-domainwhoseanalysisrequiresthis
separate chapter. If the same A-stable time-discretisation is used both in the
interior (bounded)domainand the exterior(unbounded)domain,the analysis
oncethe correctcontinuoussystem isintroducedfollowsthe same stepsasin
Chaps. 2 and 4. However, in practice it is of interest to use a coupling of an
explicit (in the interior) and implicit (in the exterior) time-stepping schemes.
Theanalysisofsuchacouplingrequiresadifferentapproachdescribedinthis
chapter.
8. Chapter 8 is an exception to the rest of the book as it addresses parabolic
problems. The main reason for including this chapter was to describe the
importantfastandobliviousalgorithmsincludingsomerecentadvancesinthe
area.AgainthefastalgorithmsareimplementedandillustratedinMatlab.
9. Finally,inChap.9,wegivethemainideasbehindtheapplicationofdata-sparse
techniques.Thisisalargeandcomplexresearchareathatwoulddeserveabook
on its own. While expertswould be able to implement the ideas described in
this chapter, for the rest of the readers we hope that this chapter manages to
effectivelydescribethemainideasandconcerns.Inanumberofplacesinthe
book, we have also concernedourselveswith the qualitativebehaviourof the
numericalsolution;in particular,we were concernedwith the dissipation and
dispersionofA-stableCQschemes.Weendthechapterbydescribingarecently
introducedmodifiedCQschemespeciallydesignedtoavoidtheseinaccuracies.
Whilethemethodsucceedsinthis,weexplainthatthetoolboxofCQanalysisis
nolongeravailableandthatthedata-sparsecompressionbecomesmoredifficult
touse.
10. The Appendix contains an eclectic mix of results and definitions that we use
throughoutthebookandthatareusuallyasubsetofanappliedmathematician’s
toolbox.
Iendthis(first)prefaceonapersonalnote.Unusually(atleastforafirstedition)
thisbookhastwoprefaces:thisone,writtenaftertheuntimelydeathofthesecond
authorandafterthecompletionofthebook,andthesecond,writtenatthestartof
thewritingofthebook.Afteranumberoffalsestarts,wefinallyhadtherightidea
fortheformatandin 2018beganwritinginearnest.Notlongafterthis, Francisco
received his diagnosis and passed on the shocking news. We continued writing,
initially at a reasonable pace but soon our weekly Skype meetings were more an
opportunity to see and chat to my friend than work on the book. Francisco kept
his spirits up remarkably well until the very end and produced a slew of high-
qualitypapersandtwobooks!Afterwards,I founditdifficultto continueworking
on the book, and there was another long break from writing it. A big thank you
for pushing me on go to our editor Rémi Lodh, and to Silvia Falletta and Letizia
x Preface
ScuderiforinvitingmetoteachanonlineExcellencecourseatPolitecnicodiTorino
in March/April2021.Preparingandteachingthiscourseandcommunicatingwith
theparticipantswereessentialinhelpingmeunderstandhowtopresentcrucialparts
ofthetheory.SomeofthematerialinChaps.2,3and4isfromthelecturenotesI
prepared for a two-week PhD course taught in 2014 at TU Vienna and organised
by Jens Markus Melenk. I am very grateful to Dugald Duncan, Penny Davies,
Matteo Ferrari, Christian Lubich, Alexander Rieder, María López-Fernández and
the anonymous referees for commenting on parts of the book and particularly to
MarkusMelenkforgivingadvicethatimprovedthepresentationsignificantly.
MyviewofthetopicsinthisbookhasbeenmoststronglyinfluencedbyChristian
Lubich, by his works and by our continuedcollaborationthat has been as fruitful
as it has been enjoyable. I will always be thankful to Wolfgang Hackbusch for
introducingmetoboundaryintegralequations(andtheirfastsolution)andtoStefan
Sauterforintroducingmetotheirtime-domaincounterparts.Inrecentyears,María
López-FernándezandI haveina series ofpapersdevelopedideasonfastsolution
of parabolic and some hyperbolic problems that inform much of the material in
Chap.8.Someofthehistoryoftime-domainboundaryintegralequationsandtheir
numericalsolutionsaregivenintheprefacebelow.
The book is now complete,unfortunatelymostly in my own voice. Those who
knewFrancisco/Javier/Panchowillenjoyhearinghisvoiceandinimitablestyleand
wit in the preface below. Those who haven’t been that lucky will enjoy getting a
smalltasteofit.
Edinburgh,UK LehelBanjai
June2022
Time-domainboundaryintegralequations(TDBIE)are integralrepresentations
of the solution of many transient problems in the physical sciences. They are
the time-domain counterpart of the very well-established collection of Boundary
IntegralEquations(BIE)forsteady-statePartialDifferentialEquations(PDE).Asit
happenswithBIE,theproblemsthatcanbeaddressedwithintegralrepresentations
is limited by (a) the need of a fundamentalsolution (which in the case of TDBIE
can be given in its transient form or through its Laplace transform, depending on
whichmethodwillbeusedfortheirdiscretisation)and(b)theimpositionofacertain
amountofhomogeneityinright-handsidesoftheequationsandininitialconditions.
Concentrated-in-space right-hand sides can sometimes be handled (not cheaply)
withvolumeintegrals,butthespiritofBIEisthatofhomogeneousequationswith
non-homogeneousand often complicatedboundaryor transmission conditions.In
the case of the time domain, TDBIE incorporate memory terms (delays or full
accumulation of the past behaviour) which make non-vanishing initial conditions
incompatible with integral representations. Similarly to non-vanishing right-hand
sidesinthe associatedPDE,theycanbehandledseparatelyusingvolumeintegral
representations.Whiletheseconditions(especiallyhavingafundamentalsolution)
seem quite restrictive, they are met by several highly relevant equations, like the
heat equation, the (acoustic) wave equation with or without damping terms, the
